Yes. The difference in surface speed for places at different latitudes can be in the order of hundreds of km/hr. This means that, in the time taken for a plane journey, the final destination may have 'moved' a significant distance from where it started, relative to the journey start point.These two animations are worth seeing. As with tidal flow (in ships) and winds, you need to plan ahead and point yourself at where your destination will be at the end of your journey. Any other course will involve a longer trip and more fuel will be used.

Not the way that webpage shows, no. The atmosphere rotates with the earth, so that pretty much takes care of any direct influence the coriolis effect would have (since the plane moves within the atmosphere, not within a fixed inertial frame independent of the earth's rotation). The Coriolis effect does strongly influence wind currents though, so the aircraft does need to factor those in. In that way, it has a significant indirect effect.

Not the way that webpage shows, no. The atmosphere rotates with the earth, so that pretty much takes care of any direct influence the coriolis effect would have (since the plane moves within the atmosphere, not within a fixed inertial frame independent of the earth's rotation). The Coriolis effect does strongly influence wind currents though, so the aircraft does need to factor those in. In that way, it has a significant indirect effect.

You are certainly right, qualitatively but I think the effect of the moving atmosphere must depend upon the amount of lateral windage of the aircraft. If there were no drag, there would be no 'coupling' to the aircraft and the effect would be zero; it would then be, as you put it, an inertial frame. The extreme case would be in a slow airship where the movement of the atmosphere would totally dominate. Whether the effect of what would be equivalent of a beam wind on a 747 for, say, an hour would cancel the coriolis force, isn't clear to me. The jet stream is enough to make a difference to flight times and courses, so it may well be
I just spent some time googling and I think you are probably right about this. The courses plotted for long haul aircraft are curved (on a chart) because they follow a great circle path. That has added to the confusion.
I couldn't find, anywhere, a reference to pilot training instructions which include coriolis in the context of navigation, so that clinches it for me about commercial flight. Otoh, I couldn't find the statement "coriolis force is not relevant to navigation" - not that that proves anything.
The situation is different for ICBMs and probably for very high altitude hypersonic craft, which are on the very atmospheric fringe.

Absolutely. Coriolis will have a significant effect on the trajectory of long-range artillery and ballistic missiles, since they are far less coupled to the atmosphere than aircraft.

The actual targeting error for shells is comparatively small, because the flight times are relatively short. In a 2 minute artillery shell flight, the error can only be a few tens of metres. Very relevant when you have a specific target, of course.This link suggest the effect could be several hundred km for an ICBM, aimed at Moscow. (A chilling article!!)

I think what happens in an airplane, is that the entire effect gets canceled, because the pilot tries to keep fly the plane in such a way that the plane seems to be horizontal on the artificial horizon.

Suppose you have a plane, flying from the equator, straight north to the northpole. The nose of the plane points north all of the time. There will be a coriolis acceleration of 2 v x ω to the right of magnitude

[tex] 2 v \frac {2 \pi}{86400} sin \theta [/tex]

where v is the speed of the plane and θ the latitude.
If v is 250 m/s , the maxium acceleration (at the northpole) is 0.036 m/s^2, or about 270 times smaller than the acceleration of gravity.

Now a ballastic missile outside of the atmosphere would just keep on accelerating, and the error in position would grow quadratically, but in an airplane the rightward speed would grow until the force of the air resistance produced enough force in the direction to the left, to produce an acceleration that canceled the coriolis acceleration.
The air resistance to sideways movement will be much higher when the plane is flying at 250 m/s, then when it is standing still btw.
I think the plane would still have a constant sideways speed that produce a course error that would get notice after a few hours.

BUT...

What would the pilot (and the artificial horizon) feel.
You can't feel the coriolis acceleration, because it's not a real acceleration, but just the plane trying to go in a straight line, but you CAN feel the acceleration caused by the air resistance, and it will would feel as if the plane is constantly pushed to the left by the air (wich in fact it is). Objects in the plane would tend to get pushed to the right and a pendulum would hang towards the right side of the plane by 1/270 radians or 0.2 degrees.

The pilot would think that the plane was banked to the right, and would bank the plane to the left until the sideways air resistance force disappeared, and a pendulum would hang straight again. This produces an acceleration to the left (because the force of the lift now points a little to the left) that cancels the coriolis acceleration.
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I guess it has to be true that, if the plane is not flying in a straight line, there must be a centripetal acceleration. It will be the continual lateral pressure of the surrounding air (with either increasing of decreasing speed, as you go north or south) that will force the plane to follow the 'right' course, rather than the 'coriolis course'.
I am not sure that the pilot would (or even would need to) alter the trim. It would just be a matter of keeping to the required course over ground (COG), which the surrounding air is pretty well forcing him / her to, in any case. This "air resistance" is only there because of the air speed gradient with latitude and would not be present if the plane were stationary. It's an 'AC coupled' thing and the extent of the lateral effect will depend on the aerodynamics of the plane's body. Are we not making assumptions that the lateral force at cruising height is still enough to cancel coriolis completely? Maybe there is a height (air density) at which there is a transition from ballistic conditions to flight conditions. How knife-like can a fuselage be made to side-slip, through the air yet still achieve lift?

The Coriolis effect is responsible for prevailing winds. For transatlantic flights, for example, that usually means eastbound flights are about one hour shorter than westbound flights. And that is recorded in timetables, so, like it or not, the Coriolis effect is factored into flying.

The direction of prevailing winds depends on latititude (link to wiki article below). The ideal path for aircraft would be a great circle path, taking prevailing winds and jetstreams into account, avoiding headwinds, and taking advantage of tailwinds.

Here is a concrete case to consider: A non-stop, great-circle flight from Denver --> New Delhi. If you plot this route out on a globe, you'll see it takes you right over the North Pole. What bearing do you fly to stay on a straight arc heading toward the North Pole? It's a 12 hour flight. You take off at 6 pm due North. By 7 pm., due to Earth's rotation has moved Denver 15 degrees Eastward from where it was when you took off, so a straight arc from 7 pm-Denver to the North Pole is 15 degrees more Westward than the direction you started along. This suggests that you want to be very slowly curving West from the local due-North, enough so that over a 12-hour flight, you've
changed direction 180 degrees. So the correction I get is 15 degrees / hour westward from due North, or 0.25 degree per minute. As pilot flying manually, I would
set a compass bearing of 359 degrees flying level. After 4 minutes, I would see that the compass has "drifted" to show a bearing of 0 degrees, and I would turn the
direction back to 359, and just keep repeating the cycle every 4 minutes. This coriolis correction keeps the tail pointed toward Denver as the Earth rotates, and
the nose pointed toward the North Pole, assuring I stay on a shortest path route. After crossing the North Pole, I just keep making the same bending correction, as
my target of New Delhi is rotating leftward in front of me. Does this sound right?
If the atmosphere didn't rotate with the Earth, I could just take off and hover (circle?) for 12 hours while New Delhi rotated around 180 degrees around the Earth's spin axis.
But the atmosphere does rotate with the Earth, so I have to fly over the pole to plow through the shortest atmospheric path.

Suppose you have a plane, flying from the equator, straight north to the northpole. The nose of the plane points north all of the time.

Wrong! You always steer "into the wind". Wind vector triangles are plotted to determine what heading must be steered in order to follow a particular course.
I haven't had a chance to read all of the foregoing posts, and won't for a while (too many numbers for me to handle right now), but I can say that when I was flying Coriolis didn't enter into it from the pilot's side. That was up to the meteorologists.

Willem2 - Here's another thought experiment about flying due North in the northern hemisphere without any wind: At any given latitude, the atmosphere and the aircraft have inertia. Because the Earth is rotating, there is some component of inertia pointing out the right side of the aircraft. Now move north from there a few kms. You're now at a higher latitude, and the radius of Earth rotation is less than before. Like an ice skater pulling in her arms to spin faster, the airplane's sideways inertia will bend its flight path slightly to the right. You have to correct for this coriolis effect by compensating with a tiny bit of left steerage to stay on course.

Pilots wouldn't have to know anything about this effect to remain on course. If they dial in a compass bearing on auto-pilot, the software will steer the plane to stay on the bearing. If flown manually, the pilot will steer the plane to keep the compass on the desired bearing. You wouldn't notice the Coriolis effect unless you were flying northward, and told auto-pilot to fly level straight-ahead (no turning). You would see the compass bearing slowly drift eastward, at about 1 degree every 4 minutes. The conservation of rotational inertia insists on this bending.